Revenge tends to cause problems, as people reciprocate back and forth. And killing the dog will not bring those birds back to life. So no, I do not think there is any right to revenge.
But since some birds were still alive, shooting the dog can help protect them. And the birds that were loose are not the only birds OP has. There are other birds in pens, and other birds that OP may want to let loose in future-- so there are a large number of birds that can be protected by removing the dog that is already proven to kill chickens. (Well, the other dogs need to get removed or confined too, but if shooting one will cause the owner to confine them all, then that is also successful in protecting the chickens.)
So I think the shooting was sensible, but only when viewed as protecting living chickens, not as revenge.